Let be a specification, that is, a consistent family of finite-volume conditional Gibbs measures for a finite-range Hamiltonian . Suppose that is the corresponding infinite-volume Gibbs measure, let be translation invariant. Given a subset of cardinality and a boundary condition , let be the random walk Metropolis chain for . It was shown by the second author and A. F. M. Smith [Stochastic Processes Appl. 49, 207-216 (1994; Zbl 0803.60067)] that converges weakly to as . The behaviour of the algorithm as is studied. In particular, choose the proposal variance . Under suitable assumptions on and it is proven that converges weakly as to an infinite-dimensional diffusion on a Hilbert space . The measure is given by , solves the equation
driven by a Brownian motion in and with the initial condition in law. The coefficient is defined in terms of the second derivative of the Hamiltonian .